Unit 10: Confidence Intervals

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 Practice Questions

1. Three things influence the margin of error in a confidence interval estimate of a population mean: sample size, variability in the population, and confidence level. For each of these quantities separately, explain briefly what happens to the margin of error as that quantity increases.


2. . A survey of 1000 Californians finds reports that 48% are excited by the opportunity to take a statistics class. Construct a 95% confidence interval on the true proportion of Californians who are excited to take a statistics class. Be sure to state/check assumptions.

3. Since your interval contains values above 50% and therefore does find that it is plausible that more than half of the state feels this way, there remains a big question mark in your mind. Suppose you decide that you want to refine your estimate of the population proportion and cut the width of your interval in half. Will doubling your sample size do this? How large a sample will be needed to cut your interval width in half? How large a sample will be needed to shrink your interval to the point where 50% will not be included in a 95% confidence interval centered at the .48 point estimate?

4. A random sample of 67 lab rats are enticed to run through a maze, and a 95% confidence interval is constructed of the mean time it takes rats to do it. It is [2.3min, 3.1 min]. Which of the following statements is/are true? (More than one statement may be correct.)

(A) 95% of the lab rats in the sample ran the maze in between 2.3 and 3.1 minutes.
(B) 95% of the lab rats in the population would run the maze in between 2.3 and 3.1 minutes.
(C) There is a 95% probability that the sample mean time is between 2.3 and 3.1 minutes.
(D) There is a 95% probability that the population mean lies between 2.3 and 3.1 minutes.
(E) If I were to take many random samples of 67 lab rats and take sample means of maze-running times, about 95% of the time, the sample mean would be between 2.3 and 3.1 minutes.
(F) If I were to take many random samples of 67 lab rats and construct confidence intervals of maze-running time, about 95% of the time, the interval would contain the population mean. [2.3, 3.1] is the one such possible interval that I computed from the random sample I actually observed.
(G) [2.3, 3.1] is the set of possible values of the population mean maze-running time that are consistent with the observed data, where “consistent” means that the observed sample mean falls in the middle (“typical”) 95% of the sampling distribution for that parameter value.


5. Two students are doing a statistics project in which they drop toy parachuting soldiers off a building and try to get them to land in a hula-hoop target. They count the number of soldiers that succeed and the number of drops total. In a report analyzing their data, they write the following:

“We constructed a 95% confidence interval estimate of the proportion of jumps in which the soldier landed in the target, and we got [0.50, 0.81]. We can be 95% confident that the soldiers landed in the target between 50% and 81% of the time. Because the army desires an estimate with greater precision than this (a narrower confidence interval) we would like to repeat the study with a larger sample size, or repeat our calculations with a higher confidence level.”

How many errors can you spot in the above paragraph?


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